Contact Information
Department of Mathematics
University of British Columbia

Office: Math 126

Current TA Duty:
Math 320: Real Analysis I

E-mail: yezichun [at] math.ubc.ca

I am a Ph.D. student in Mathematics at University of British Columbia, where I am part of the probability group, and my supervisor is David Brydges. My research interests include probability theory and statistical mechanics.

Here is my CV.

Before becoming a Phd, I got my master degree in May 2012 from UBC, under the supervision of David Brydges.

Research

1. Scalar field

The theory of convex functions of the gradient of a scalar field is well developed and applies to problems arising from anharmonic crystals, dipole gases, and random surfaces. Let $$\phi_j \in \mathbb{R}$$ with $$j \in \Lambda \subset \mathbb{Z}^d$$. In my research we consider a class of models described by finite volume measures which take the form $\text{(Normalisation)}e^{-\sum\limits_j(V(\bigtriangledown\phi_i)+\epsilon\phi_j^2)}\prod\limits_{j \in \Lambda}d\phi_j.$ where $$\epsilon >0$$ and for $$\bigtriangledown\phi$$ large, $V(\bigtriangledown\phi_i) = O\big((\bigtriangledown\phi_i)^{2\alpha}\big).$ We are especially interested in large deviations of the field, given by $$e^{\gamma\phi_0}$$. When $$\alpha = 1/2$$, David Brydges and Thomas Spencer[1] gave a uniform bound of $$e^{\gamma\phi_0}$$ in $$\epsilon$$ and $$\Lambda$$. My work is to find a simila bound for $$\alpha \in (0, 1)$$. I prove that for $$\alpha < 0.5$$, $$e^{\gamma\phi_0}$$ blow up when $$\epsilon \to 0$$, and we believe that a bound exist for $$\alpha > 0.5$$, i.e. there is a phase transition at 0.5. This work are still in process.

2. Dimer system:

A monomer-dimer system is de ned by a graph G = (E; V ) and a set of dimers (covering an edge with its two vertices) and monomers (covering one vertex) on G. The dimers and monomers have exclusion interaction and thus satisfy the condition of compatibility.We can use the Mayer expansion to give a expression about the pressure of the monomer-dimer system. However, this expansion need the conditions of low density to be convergent.

In my master essay
[2], I use two method to deal with convergence issue of the Mayer expansion.
• To study the remainder of the Mayer expansion like Peano remainder of the Taylor series. For the the rst order remainder, we prove that it can be expressed as apartition function in which order of the weights of molecules are larger than 1 and give a approximate formula to estimate it. we also prove a similar proposition for the second order remainder.

• To study the location of the zeroes of the partition function. We use the Lee-Yang theorem to prove that the zeroes of partition function are all on the negative real axis. So we rove a better bound for the convergence radius of the Mayer expansion and show the Mayer expansion in right half plane will converge.

In addition to my essay, I also try to simulate dimers system in matlab and compare the results to theoretical order estimate.

Past TA Duty